The clustering properties of local, S1.4 GHz ≥ 1 mJy, radio sources are investigated for a sample of 820 objects drawn from the joint use of the Faint Images of the Radio Sky at 20 cm (FIRST) and 2dF Galaxy Redshift surveys. To this aim, we present 271 new bJ ≤ 19.45 spectroscopic counterparts of FIRST radio sources to be added to those already introduced in our previous paper. The two-point correlation function for the local radio population is found to be entirely consistent with estimates obtained for the whole sample of 2dFGRS galaxies. From measurements of the redshift-space correlation function Θ(s) we derive a redshift-space clustering length s0 = 10.7-0.7+0.8 Mpc, while from the projected correlation function ζ (rT) we estimate the parameters of the real-space correlation function ζ (r) = (r/r0) -γ, r0 = 6.7-1.10.9 Mpc and γ = 1.6 ± 0.1, where h = 0.7 is assumed. Different results are instead obtained if we only consider sources that present signatures of active galactic nucleus (AGN) activity in their spectra. These objects are shown to be very strongly correlated, with r0 = 10.9-1.2+10 Mpc and γ = 2 ± 0.1, a steeper slope than has been claimed in other recent works. No difference is found in the clustering properties of radio-AGNs of different radio luminosity. Comparisons with models for ζ (r) show that AGN-fuelled sources reside in dark matter haloes more massive than ∼1013.4 M⊙, higher than the corresponding figure for radio-quiet quasi-stellar objects. This value can be converted into a minimum black hole mass associated with radio-loud, AGN-fuelled objects of M BHmin ∼ 109 M⊙. The above results then suggest - at least for relatively faint radio objects - the existence of a threshold black hole mass associated with the onset of significant radio activity such as that of radio-loud AGNs; however, once the activity is triggered, there appears to be no evidence for a connection between black hole mass and level of radio output.